A Curious Congruence Involving Alternating Harmonic Sums

TL;DR

This paper proves a novel congruence involving alternating harmonic sums over integers coprime to a prime p, relating it to Bernoulli numbers and powers of p, expanding understanding of harmonic sums in modular arithmetic.

Contribution

The paper establishes a new congruence involving alternating harmonic sums and Bernoulli numbers for prime powers, which was previously unexplored.

Findings

Proves a congruence involving sums over coprime integers and Bernoulli numbers.

Connects alternating harmonic sums to prime powers in modular arithmetic.

Provides a new tool for analyzing harmonic sums in number theory.

Abstract

Let $p$ be a prime and ${\mathcal{P}_{p}}$ the set of positive integers which are prime to $p$. We establish the following interesting congruence \[\sum\limits_{\begin{smallmatrix} i+j+k={{p}^{r}} i,j,k\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{{{(-1)}^{i}}}{ijk}}\equiv \frac{{{p}^{r-1}}}{2}{{B}_{p-3}}\, (\bmod \, {{p}^{r}}).\]